Find the values of x, y and z, so as to satisfy the equations given below: 5x – 3y + 7z = 22; 3x – 5y – 2z = – 46; 2x – 2y + 5z = 24

Correct — C. All four triples are built from the same numbers 5, 3 and 8 with different signs, so a single equation screens them.

Put each triple into 5x − 3y + 7z = 22:

(−5, 3, −8) → −25 − 9 − 56 = −90
(5, −8, 3) → 25 + 24 + 21 = 70
(−5, 3, 8) → −25 − 9 + 56 = 22
(−5, −3, 8) → −25 + 9 + 56 = 40

Check the survivor in the other two equations:
3(−5) − 5(3) − 2(8) = −15 − 15 − 16 = −46
2(−5) − 2(3) + 5(8) = −10 − 6 + 40 = 24

All three equations hold for x = −5, y = 3, z = 8 → option (c)

Why the others are wrong

• (a)
  x = – 5, y = 3, z = – 8 — Only the sign of z differs from the solution. With z = −8 the first equation gives −25 − 9 − 56 = −90, nowhere near 22.
• (b)
  x = 5, y = – 8, z = 3 — This triple makes x positive and swaps the values of y and z. The first equation then gives 25 + 24 + 21 = 70.
• (d)
  x = – 5, y = – 3, z = 8 — Only the sign of y differs. The first equation gives −25 + 9 + 56 = 40, and the third fails too: 2(−5) − 2(−3) + 5(8) = 36, not 24.

Three linear equations in three unknowns are solved by elimination: remove the same variable from two different pairs, which leaves two equations in two unknowns, then work back.

Done here, 2 × (first) + 7 × (second) removes z and gives 31x − 41y = −278, while 5 × (second) + 2 × (third) removes z and gives 19x − 29y = −182. That pair yields y = 3 and x = −5, and the third equation then gives z = 8.

When the options are complete triples, substitution is faster. Elimination is what you need the moment an option reads no solution or the answer is asked as a single expression.

Key facts

• A triple is a solution only if it satisfies every equation in the system.
• Eliminating z from the first two equations here gives 31x − 41y = −278.
• The system has the unique solution x = −5, y = 3, z = 8.

Common traps

• Substituting into one equation and marking, without checking the other two.
• Sign slips on the 7z and −3y terms, which is exactly what separates three of these options.
• Reading the semicolons as one long expression rather than three conditions.

No directly related past PYQ was found.